Viral evolution: beyond drift and shift

•Sequencing enables exploration of viral evolution at unprecedented scales.•New quantitative methods have emerged from sequencing-enabled studies.•Novel tools, some from statistical physics, are needed to quantify virus evolution.•Information theory and topology can be used to quantify virus segment reassortment.

Technological advances have allowed aspects of viral evolution to be explored at unprecedented scales. As a consequence, new quantitative approaches are needed to investigate features of viral evolution that fall outside traditional areas of study, such as antigenic evolution. We examine three areas of viral evolution where tools from disciplines such as statistical physics, topology, and information theory have been used recently as quantitative frameworks for large-scale studies and, in some cases, suggest a novel theoretical approach to a problem. Ongoing interaction among these disciplines with biology is necessary so that experimental researchers can determine which quantitative tools are right for them and quantitative researchers can learn which aspects of viral evolution can be understood and advanced with their approaches.